A moderated theorem was established in this study to derive an infinite series as an absolute Riesz | Ñ,P^α_n;ð;ý |_k summable by applying an almost increasing sequence. Suitable conditions were applied for obtaining some well-known applications of the presented theorem. It was observed from the investigation that absolute summability could be used for Bounded Input Bounded Output stability and to bind the output of the signals as absolute summable, it was a necessary and sufficient condition for BIBO stability.